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Ravi Vakil, an Associate Professor in the Department of Mathematics at Stanford University ponders a students solution to a math problem. Vakil who, according to his website, each fall coordinates the William Lowell Putnam competition at Stanford, and in conjunction with that runs a weekly seminar for talented undergraduates, as well as a problem-solving Masterclass for experts. This year the Sixty-Seventh Annual William Lowell Putnam Mathematical Competition will be held on Saturday, December 2, 2006 and is administered by The Mathematical Association of America.
Chris Stewart / The Chronicle Ravi Vakil, Stanford, William Lowell Putnam Mathematical Competition Ran on: 12-02-2006
Associate Professor Ravi Vakil ponders a student's solution to a math problem.

Photo: Chris Stewart

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Ravi Vakil, an Associate Professor in the...

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Ravi Vakil, an Associate Professor in the Department of Mathematics at Stanford University is pleased with a student's math solution. Vakil who, according to his website, each fall coordinates the William Lowell Putnam competition at Stanford, and in conjunction with that runs a weekly seminar for talented undergraduates, as well as a problem-solving Masterclass for experts. This year the Sixty-Seventh Annual William Lowell Putnam Mathematical Competition will be held on Saturday, December 2, 2006 and is administered by The Mathematical Association of America.
Chris Stewart / The Chronicle Ravi Vakil, Stanford, William Lowell Putnam Mathematical Competition Ran on: 12-02-2006
Associate Professor Ravi Vakil ponders a student's solution to a math problem.

Aman Kumar spends Monday nights in Stanford's math department plunging into theorems for fun. He gets no credit for this. It sucks up precious time. But like dozens of other students, Kumar is there because of Ravi Vakil.

In the universe of Stanford math junkies, Vakil has a glint of rock star. A couple of his mathematical proofs have garnered international attention, and he's a legend in the world of math competitions.

The 36-year-old associate professor has almost single-handedly revived at Stanford the world's most difficult collegiate math contest -- the William Lowell Putnam Mathematical Competition, which takes place Saturday at Stanford. It's the pinnacle of college math competitions, one so hard and so prestigious the winner receives a full scholarship to Harvard for a doctorate degree.

Scoring a single point on this 120-point exam is a feat. Maybe that's why Vakil inspires awe. All four years as an undergraduate at the University of Toronto, he was a "Putnam fellow" or one of the top five annual performers, all of whom achieved extremely high scores. Winning all four years is a distinction held by Vakil and only five others since the first Putnam competition in 1938.

"When you learn the Putnam with Ravi, math is a fun, sexy thing rather than a boring, repetitive, learn-by-rote thing," said Kumar, a sophomore. "Ravi just represents this dynamic, creative math rather than staid and stagnant math."

Vakil's field is algebraic geometry, which he said requires knowledge in many mathematical areas just to "get in the front door." High schoolers might get a taste of it by, say, creating an algebraic equation to show where a line meets a circle.

"This is that turned on its head and through the looking glass," he said of his field. It's basically "really hard problems in algebra. I turn them into pictures in my mind. I can move them around and see how they fit together."

He has written two books, helped write another and has 500 pages of notes for a fourth. This is critical for professors who expect to advance their careers in the upper echelons of academia. But he also advises graduate students and throws himself into his classes, which won him the dean's award for distinguished teaching two years ago. And then there are the many hours he voluntarily coaches students for the Putnam.

On a recent Monday night, students sat in room 383N eating pizza, drinking soda and watching classmates outline proofs on a chalkboard for problems like this:

Five points lie in an equilateral triangle of size 1. Show that two of the points lie no farther than 1/2 apart. Can the " 1/2" be replaced by anything smaller? Can it be improved if the "five" is replaced by "six"?

"I can't necessarily tell you off the top of my head how to solve them," Vakil told the group. "But I can tell you how to think about them."

Some of the students seemed self-conscious. One excused his proof ahead of time, explaining, "I haven't had time to think about it in an elegant way." But after the student presented his work on the chalkboard -- a solution that resembled hieroglyphics -- Vakil didn't think so.

"He's using something that's good to use -- he's using a generating series for the binomial coefficient," Vakil told everyone. "There are other ways of doing it, but this is a particularly elegant way of getting there."

Students who take the eight-hour Putnam, he said, do so for the same reasons people climb Mount Everest: because it's there.

"These are the best people on the planet -- that attitude," he said. "It's really hard, and only a few people can handle it."

Vakil has long liked math competitions. The Toronto Star featured him at age 18 in a news story headlined "Etobicoke math whiz strikes gold in Australia" about his International Mathematical Olympiad performance. He grew up with academic parents -- his father was a professor of preventative medicine at the University of Toronto, his mother a high school math teacher -- but they never pushed him at math. For a time he studied economics and political science, with the goal of becoming an ambassador but realized those people don't wield the influence he had initially hoped. After graduating from the University of Toronto, he earned his doctorate in mathematics at Harvard University and while there, secretly wrote a book (he was supposed to be working full time on his dissertation) that explains how to think about math through vignettes.

He doesn't like writing, he said, but the books well up within him: "This book has to be written, and I have no choice but to write it."

His colleagues marvel at all he can do in a day. Others have taken notice of his accomplishments, too. It's common knowledge other universities are courting Vakil. Stanford, hoping he will stay, is promoting him to full professor. One colleague called him a genius. Another said he's almost too generous with his time. Math department Chairman Yakov Eliashberg said, "He's fast and smart and at the same time, doing very deep things."

While Vakil hasn't shattered the math world with the best proof of the last 100 years, he has come up with a couple of blockbusters that have generated buzz. His latest one, "Murphy's Law," in a nutshell says that deforming a curve in space or another geometric object -- a concept mathematicians have wrestled with for more than a century -- is not always governed by a simple, beautiful equation, as many have hoped. Vakil proved all equations can devolve into horrible nastiness. The paper, he said, saddened those who see the world through a mathematical lens.

"Anytime you say the world is a cruel and unforgiving place," Vakil said, "that doesn't make you happy."

Students, knowing he might leave, have signed up for "Modern Algebra II" this winter because they fear it might be one of the last chances to take a Vakil class, said sophomore Katie Howard. She knows his talent from the Monday night Putnam study sessions. She has received the mass weekly e-mails with practice problems and kindly statements like "If you have any questions (about any of the above, or about life in general), just let me know."

"He's one of the most fantastic math teachers I've seen," she said. "He seems to care about everyone understanding what he teaches. He will work with you until you understand."

That's the main reason he volunteers as a Putnam coach: He wants to push students to understand and think about math creatively.

He is betting Stanford will have its best Putnam year. Last winter, 37 Stanford students scored in the top 500 out of the 3,500 nationwide who take the exam. By that measure, it put the university third, behind MIT (66) and Harvard (40). Each year, the Putnam generates more excitement at Stanford. Random students are e-mailing Vakil saying they want to take it. All are welcome. Sometimes smart students who know the least can do surprisingly well: They're fearless and their minds are open to possibilities, said Vakil. If you know too much -- if you have a hammer -- everything looks like a nail.

On Monday, the last study session of the year, Vakil was suffering from a terrible bug that hit him so hard that he had postponed a trip to Chicago to present his work. But he couldn't abandon his students in the final push before the Putnam. He can't imagine not coordinating the competition for Stanford.

"I'm here because a lot of people older than me volunteered for things like this because they thought it was important to do."